Probability: chances of winning

1. The problem statement, all variables and given/known data
Consider a game where a player rolls 2 dice. The player looses if the sum of the dice of the first roll is 2, 3 or 12, and wins if the sum of the dice is 7 or 11. If the outcome of the first roll is 4, 5, 6, 8, 9 or 10, the player continues to roll the dice until the initial outcome is rolled again (player wins) or the 7 appears (the player looses).
Find the probability of winning the game.

3. The attempt at a solution
The size of the sample space is |[tex]\Omega[/tex]| = [tex]6 * 6 = 36[/tex], since we have two six-sided dice, so a total of 36 possible outcomes on a given throw.

I gather that if the player rolls a sum of 2, 3, or 12 on their first try, they automatically lose the game. Therefore, I let event [tex]E_{L}[/tex] denote the event that either of those sums is thrown. Therefore, [tex]E_{L}[/tex] is the set of all outcomes that result in one of those sums.

I also worked out the size of events [tex]E_{4}[/tex], [tex]E_{5}[/tex], [tex]E_{6}[/tex], [tex]E_{7}[/tex], [tex]E_{8}[/tex], [tex]E_{9}[/tex], [tex]E_{10}[/tex], where each subscript represents the number of outcomes that could lead to that sum being thrown. I got: